Branching rules for the Weyl group orbits of the Lie algebraA_n

Abstract

The orbits of Weyl groups W (An) of simple An type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of An. Matrices transforming points of the orbits of W (An) into points of subalgebra orbits are listed for all cases n ≤ 8 and for the infinite series of algebra-subalgebra pairs An ⊃ A n−k−1 × A k × U 1 , A 2n ⊃ Bn, A 2n−1 ⊃ Cn, A 2n−1 ⊃ Dn. Numerous special cases and examples are shown.

LieART—A Mathematica application for Lie algebras and representation theory

“…because in the former case the negative roots are included in W T ξ i , i.e., the −1, while only positive roots are in the orbit W T ξ i if ξ i / ∈∆ T , requiring a factor of 2 for the same reason as on the left-hand side of (23)…”

“…Our matrices are correct and consistent, but one may find different projection matrices in the literature, which are also correct. (An extensive collection of projection matrices can be found in [23] for the Lie algebra A n and in [24] for the Lie algebras B n , C n and D n .) Once a projection matrix is known it can be used for the decomposition of all irreps of the algebra-maximal-subalgebra pair.…”

Centralizers of maximal regular subgroups in simple Lie groups and relative congruence classes of representations

“…When dealing with large-scale computation for representations, one often needs to break down the problem into smaller ones for individual orbits. Orbit-orbit branching rules are computed with the projection matrix method for orbits of W (A n ) in [4] and for orbits of W (B n ), W (C n ) and W (D n ) in [5].…”

“…(1, 0, 0, 0, 0, 0) ⊃ (0, 0, 0, 0, 1)[1] + (1, 0, 0, 0, 0)[−2] + (0, 0, 0, 0, 0) [4] where the term in square brackets is the relative congruence class.…”

Branching rules for Weyl group orbits of simple Lie algebrasBn,CnandDn

“…The motivation for the present paper is the same as in [1]. There are four important points to note: firstly, orbit branching rules are implicitly required for the computation of branching rules of representations of the same Lie algebrasubalgebra pairs.…”